Related papers: Riemann-Hilbert Problem for the Matrix Laguerre Bi…
The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain…
This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlev\'e equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the…
The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain…
First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data.…
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us…
In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE…
In this paper we study the gap probability problem in the Gaussian Unitary Ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke…
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…
In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain…
We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of…
The D7 degeneration of the Painleve-III equation has solutions that are rational functions of $x^{1/3}$ for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We…
We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period $2$ in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel…
This article is firstly a historic review of the theory of Riemann-Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert's 21st problem and Plemelj's work associated with it. The secondary…
We find a local $(d+1) \times (d+1)$ Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree $d$.…
In the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to…
We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation…
We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the…
We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide…